System of vectors, linear dependence

1. The problem statement, all variables and given/known data
Prove that if in a system of vectors: [itex]S_a =\{a_1, a_2, ..., a_n\} [/itex] every vector [itex]a_i[/itex] is a linear combination of a system of vectors: [itex]S_b = \{b_1, b_2, ..., b_m\}[/itex], then [itex]\mathrm{span}(S_a)\subseteq \mathrm{span}(S_b)[/itex]

2. Relevant equations

3. The attempt at a solution
We know due to [itex]a_j[/itex] being a linear combination, that every [itex]a_j\in S_a = \sum\limits_{j=1}^m c_j\cdot b_j[/itex] where [itex]b_j\in S_b, c_j\in\mathbb{R}\setminus\{0\}[/itex]
But where should I go from here? Suggestions?

3. The attempt at a solution
We know due to [itex]a_j[/itex] being a linear combination, that every [itex]a_j\in S_a = \sum\limits_{j=1}^m c_j\cdot b_j[/itex] where [itex]b_j\in S_b, c_j\in\mathbb{R}\setminus\{0\}[/itex]

The ##c_j## can be zero. The span of a set S is the set of all linear combinations of elements of S, including linear combinations where one or more (maybe all) of the coefficients are zero.

Considering that a linear span is a vector space, then it is closed under multiplication with a scalar. Therefore, every [itex]a\in L(A)[/itex] implies [itex]a\in L(B)\Leftrightarrow L(A)\subseteq L(B)_{\square}[/itex]

I would just start with a simple statement like "Let ##x\in L(A)##." Then you can use the definition of ##L(A)## to say something about ##x##. This statement will involve the ##a_k##. Then you can use what you know about the ##a_k## to say something else. And so on. At some point you should be able to conclude that ##x\in L(B)##. Then you will have proved that ##L(A)\subseteq L(B)##.

Be careful with your statements. The quoted statement above is saying that every vector in the subspace ##L(A)## implies some statement. Statements are implied by other statements, not by vectors.